Poole, D. & Crowley, M., 2013. Cyclic causal models with discrete variables: Markov chain equilibrium semantics and sample ordering. In IJCAI International Joint Conference on Artificial Intelligence. Beijing, China, pp. 1060–1068. Available at: http://dl.acm.org/citation.cfm?id=2540281.
We analyze the foundations of cyclic causal models for discrete variables, and compare structural equation models (SEMs) to an alternative semantics as the equilibrium (stationary) distribution of a Markov chain. We show under general conditions, discrete cyclic SEMs cannot have independent noise; even in the simplest case, cyclic structural equation models imply constraints on the noise. We give a formalization of an alternative Markov chain equilibrium semantics which requires not only the causal graph, but also a sample order. We show how the resulting equilibrium is a function of the sample ordering, both theoretically and empirically.